Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xlimpnfmpt | Structured version Visualization version GIF version |
Description: A function converges to plus infinity if it eventually becomes (and stays) larger than any given real number. (Contributed by Glauco Siliprandi, 5-Feb-2022.) |
Ref | Expression |
---|---|
xlimpnfmpt.k | ⊢ Ⅎ𝑘𝜑 |
xlimpnfmpt.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
xlimpnfmpt.z | ⊢ 𝑍 = (ℤ≥‘𝑀) |
xlimpnfmpt.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) |
xlimpnfmpt.f | ⊢ 𝐹 = (𝑘 ∈ 𝑍 ↦ 𝐵) |
Ref | Expression |
---|---|
xlimpnfmpt | ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xlimpnfmpt.f | . . . 4 ⊢ 𝐹 = (𝑘 ∈ 𝑍 ↦ 𝐵) | |
2 | nfmpt1 5157 | . . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑍 ↦ 𝐵) | |
3 | 1, 2 | nfcxfr 2975 | . . 3 ⊢ Ⅎ𝑘𝐹 |
4 | xlimpnfmpt.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | xlimpnfmpt.z | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | xlimpnfmpt.k | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
7 | xlimpnfmpt.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℝ*) | |
8 | 6, 7, 1 | fmptdf 6876 | . . 3 ⊢ (𝜑 → 𝐹:𝑍⟶ℝ*) |
9 | 3, 4, 5, 8 | xlimpnf 42115 | . 2 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑘))) |
10 | nfv 1911 | . . . . . 6 ⊢ Ⅎ𝑘 𝑖 ∈ 𝑍 | |
11 | 6, 10 | nfan 1896 | . . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑖 ∈ 𝑍) |
12 | 5 | uztrn2 12256 | . . . . . . . 8 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑖)) → 𝑘 ∈ 𝑍) |
13 | 12 | adantll 712 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑖)) → 𝑘 ∈ 𝑍) |
14 | simpll 765 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑖)) → 𝜑) | |
15 | 14, 13, 7 | syl2anc 586 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑖)) → 𝐵 ∈ ℝ*) |
16 | 1 | fvmpt2 6774 | . . . . . . 7 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝐵 ∈ ℝ*) → (𝐹‘𝑘) = 𝐵) |
17 | 13, 15, 16 | syl2anc 586 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑖)) → (𝐹‘𝑘) = 𝐵) |
18 | 17 | breq2d 5071 | . . . . 5 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘𝑖)) → (𝑦 ≤ (𝐹‘𝑘) ↔ 𝑦 ≤ 𝐵)) |
19 | 11, 18 | ralbida 3230 | . . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (∀𝑘 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑘) ↔ ∀𝑘 ∈ (ℤ≥‘𝑖)𝑦 ≤ 𝐵)) |
20 | 19 | rexbidva 3296 | . . 3 ⊢ (𝜑 → (∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑘) ↔ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝑦 ≤ 𝐵)) |
21 | 20 | ralbidv 3197 | . 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝑦 ≤ (𝐹‘𝑘) ↔ ∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝑦 ≤ 𝐵)) |
22 | breq1 5062 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ 𝐵 ↔ 𝑥 ≤ 𝐵)) | |
23 | 22 | rexralbidv 3301 | . . . . 5 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝑦 ≤ 𝐵 ↔ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝑥 ≤ 𝐵)) |
24 | fveq2 6665 | . . . . . . 7 ⊢ (𝑖 = 𝑗 → (ℤ≥‘𝑖) = (ℤ≥‘𝑗)) | |
25 | 24 | raleqdv 3416 | . . . . . 6 ⊢ (𝑖 = 𝑗 → (∀𝑘 ∈ (ℤ≥‘𝑖)𝑥 ≤ 𝐵 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ 𝐵)) |
26 | 25 | cbvrexvw 3451 | . . . . 5 ⊢ (∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝑥 ≤ 𝐵 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ 𝐵) |
27 | 23, 26 | syl6bb 289 | . . . 4 ⊢ (𝑦 = 𝑥 → (∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝑦 ≤ 𝐵 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ 𝐵)) |
28 | 27 | cbvralvw 3450 | . . 3 ⊢ (∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝑦 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ 𝐵) |
29 | 28 | a1i 11 | . 2 ⊢ (𝜑 → (∀𝑦 ∈ ℝ ∃𝑖 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑖)𝑦 ≤ 𝐵 ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ 𝐵)) |
30 | 9, 21, 29 | 3bitrd 307 | 1 ⊢ (𝜑 → (𝐹~~>*+∞ ↔ ∀𝑥 ∈ ℝ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)𝑥 ≤ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 class class class wbr 5059 ↦ cmpt 5139 ‘cfv 6350 ℝcr 10530 +∞cpnf 10666 ℝ*cxr 10668 ≤ cle 10670 ℤcz 11975 ℤ≥cuz 12237 ~~>*clsxlim 42091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fi 8869 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-z 11976 df-uz 12238 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-topgen 16711 df-ordt 16768 df-ps 17804 df-tsr 17805 df-top 21496 df-topon 21513 df-bases 21548 df-lm 21831 df-xlim 42092 |
This theorem is referenced by: (None) |
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